# CA-set

The complement of an -set in a complete separable metric space ; that is, is a -set if is an -set, or, in other words a -set is a projective set of class 2. There is an example of a -set that is not an -set. Any -set is a one-to-one continuous image of some -set (Mazurkiewicz's theorem).

A point is called a value of order of a mapping if there is one and only one point such that . The values of order 1 of a -measurable mapping on an arbitrary Borel set form a -set (Luzin's theorem). The converse is true: Let be any -set belonging to a space . Then there is a continuous function defined on a closed subset of the irrational numbers such that is the set of points of order 1 of . Kuratowski's reduction theorem: Given an infinite sequence of -sets there is a sequence of disjoint -sets such that and .

#### References

[1] | K. Kuratowski, "Topology" , 1 , Acad. Press (1966) (Translated from French) |

#### Comments

A -set is also called a co-analytic set, their class is nowadays denoted by . See also -set.

**How to Cite This Entry:**

CA-set. B.A. Efimov (originator),

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=CA-set&oldid=15451